5 Answers
5

Because of the way that Cheese should be pronounced, especially when it should be pronounced emphatically (the exclamation mark at the end), it should cause your cheeks to rise and your lips to part and your eyes to slightly lower on the corners. This causes your face to lighten.

Additionally, it is rather an odd command, so it generally induces people to smile while opening their cheeks and exposing more of their face to the camera.

As "say cheese" has become the norm, fewer people actually follow through with the command, which means that photographers are tending to use various creative phrases (i.e. "Don't smile", or "Say fuzzy pickles!") to get people to laugh.
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zzzzBovMar 16 '11 at 20:17

Interestingly, while saying cheese may make you look like you are smiling, saying "gorgonzola" merely makes you look like you are pouting.
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PuzbieFeb 2 '13 at 8:23

The idea behind "say cheese" is not to make a joke, but to get your mouth into a position that resembles a smile. Depending on how you say them, ee sounds like in cheese can cause you to widen your mouth, sort of like a smile.

(Of course, unless you're being a wise-ass, you shouldn't actually say cheese, or anything else really, when you're told to "say cheese"; instead, you should just smile to the best of your ability.)